# 007B Sample Midterm 3, Problem 1 Detailed Solution

Divide the interval  $[0,\pi ]$ into four subintervals of equal length   ${\frac {\pi }{4}}$ and compute the right-endpoint Riemann sum of  $y=\sin(x).$ Background Information:

1. The height of each rectangle in the right-hand Riemann sum

is given by choosing the right endpoint of the interval.

Solution:

Step 1:
Let  $f(x)=\sin(x).$ Each interval has length  ${\frac {\pi }{4}}.$ Therefore, the right-endpoint Riemann sum of  $f(x)$ on the interval  $[0,\pi ]$ is

${\frac {\pi }{4}}{\bigg (}f{\bigg (}{\frac {\pi }{4}}{\bigg )}+f{\bigg (}{\frac {\pi }{2}}{\bigg )}+f{\bigg (}{\frac {3\pi }{4}}{\bigg )}+f(\pi ){\bigg )}.$ Step 2:
Thus, the right-endpoint Riemann sum is

${\begin{array}{rcl}\displaystyle {{\frac {\pi }{4}}{\bigg (}\sin {\bigg (}{\frac {\pi }{4}}{\bigg )}+\sin {\bigg (}{\frac {\pi }{2}}{\bigg )}+\sin {\bigg (}{\frac {3\pi }{4}}{\bigg )}+\sin(\pi ){\bigg )}}&=&\displaystyle {{\frac {\pi }{4}}{\bigg (}{\frac {\sqrt {2}}{2}}+1+{\frac {\sqrt {2}}{2}}+0{\bigg )}}\\&&\\&=&\displaystyle {{\frac {\pi }{4}}({\sqrt {2}}+1).}\\\end{array}}$ ${\frac {\pi }{4}}({\sqrt {2}}+1)$ 